Derivation of the Heat Equation

The heat equation equation is derived in the Wikipedia site. It is instructive to read this derivation.

It is based on Fourier's law which says that the heat exchanged through a surface is proportional to minus the gradient of the temperature across the surface and on energy conservation.

The equation reads

$$\frac{ \partial T}{\partial t} = \alpha \Delta^2 T = 0$$ (1)

where

$$\alpha = \frac{k}{C_p \vert rho}$$ (2)

The constants are $k$, the thermal conductivity, $C_p$, the specific heat capacity and $\rho$ the density.

$\alpha$ is called the thermal diffusivity.